The kinetic molecular theory describes how gases behave ideally. A molecule is the smallest particle of a pure substance that can exist independently. It may contain one or more atoms and their molecules may be monatomic, diatomic, and triatomic. Gas is the physical state of matter that has neither definite shape nor fixed volume.

When energy is provided to a system, it is randomly distributed i.e. collisions are perfectly elastic and kinetic energy remains conserved in ideal conditions.

The motion of ideal gas molecules in a closed container is perfectly elastic. It means that the kinetic energy remains conserved. This is the basic principle on which all the gas laws are based.

## Contributions of Scientists in KMT

- In 1738, Bernoulli introduced the kinetic molecular theory of gases.
- In 1857, Clausius derived the kinetic equation and deduced all gas laws from it.
- Maxwell, later on, introduced the law of distribution of velocities.
- Boltzman introduced the distribution of energies among gas molecules.
- Van der Waal explained the cause of deviation of matter phases from ideality.

## Postulates of kinetic molecular theory

- Gas molecules have random motion, and thus, they collide with each other and with the walls of the container.
- Collisions of gas molecules are perfectly elastic. They exert pressure on the walls of the container.
- The gas molecules are widely separated from each other. There are large spaces between them.
- The forces of attraction between the gas molecules are very low.
- The actual volume of gas molecules is negligible as compared to the volume of the gas.
- Motion of molecules due to gravity is negligible.
- Average kinetic energy is directly proportional to the absolute temperature of the gas.

K.E ∝ T°

# Equation of Kinetic Molecular Theory (KMT)

The final form of the kinetic equation is

PV = 1/3 mNc̄^{2}

where,

- P = pressure
- V = volume
- m = mass of one mole of the gas
- N = number of molecules of gas in the container
- c̄
^{2}= root mean square velocity

### Root Mean Square Velocity (c_{rms})

Maxwell introduced and explained the distribution of velocities. if there is n_{1} number of molecules with velocity c_{1} and n_{2} molecules with velocity c_{2}. Then by relating all quantities,

c_{rms} = c_{1}+c_{2}+c_{3}/n_{1}+n_{2}+n_{3}

c_{m} = c̄^{ }/ N

When we take the square root of c̄, it is called the root mean square velocity c_{rms}.

c_{rms }=√(c̄^{2})

The expression for the root mean square velocity deduced the kinetic equation is as follows;

c_{rms }= **√**(3RT/M)

where,

- c
_{rms}= root mean square velocity - M = molecule with Velocity
- T = temperature

The above equation explains the relationship between the absolute temperature and velocity of the gas molecule.

**Related resources**

## Gas Laws and Kinetic molecular theory

The kinetic molecular theory of gases explains the gas laws. In fact, all gas laws can be derived from the kinetic molecular theory of gases.

### 1. Boyle’s law

In 1662, physicist Robert Boyle formulated an empirical relation between the pressure of a given amount of a gas and its volume at a constant temperature. He stated that the kinetic energy of a gas is directly proportional to its temperature.

PV = k

### Derivation of Boyle’s law from KMT

As the kinetic energy of a gas is directly proportional to the absolute temperature (postulate no. 7 of KMT).

K.E ∝ T

The kinetic energy (1/2 mv^{2}) of N molecules is 1/2mNc̄^{2} if its mean squared velocity is used,

1/2 mNc̄^{2} ∝ T

Converting proportionality to equality,

1/2 mNc̄^{2} = kT

where,

- K is the constant of proportionality.

Now, according to the kinetic equation of gases,

PV = 1/2 mNc̄^{2}

Multiplying and dividing with 2,

PV = ⅔ (½ mNc̄^{2})

PV = 2/3 kT → equation (1)

As the temperature (T) and k are constant,

2/3 KT = k’

By constant temperature and a fixed number of molecules, the product PV is a constant quantity.

PV = k’ (Boyle’s law)

### 2. Charles law

In 1802, Joseph Louis Gay-Lussac formulated the expansion of gases under constant pressure and then credited this to the unpublished work of Jacques Charles. It stated that the volume occupied by a fixed amount of a gas is directly proportional to its absolute temperature, with pressure being constant.

V/T = k

### Derivation of Charles law from KMT

By considering derived equation (1) from the above section,

PV = 2/3 KT

Rearranging,

V = ⅔ KT / P

= 2K / 3PT

Now, at constant pressure,

2K/3P= k”

where,

- k” = constant

V = k” T

V/T = k” (Charles law)

### Avagadro’s law

In 1811, Amedeo Avogadro said that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

N_{1 }= N_{2 }

### Derivation of Avogadro’s law from KMT

Assuming a system of two gases at same pressure having same volumes, their number of molecules being N_{1} and N_{2}, the mass of molecules m_{1} and m_{2,} and their mean square velocities as c_{1} and c_{2} respectively, the relative kinetic equations shall be as,

PV = 1/3 m_{1}N_{1 }c̄^{2}_{1}

PV = 1/3 m_{2}N_{2} c̄^{2}_{2}

Comparing both equations here, we get,

1/3 m_{1}N_{1 }c̄^{2}_{1} = 1/3 m_{2}N_{2} c̄^{2}_{2}

m_{1}N_{1 }c̄^{2}_{1} = m_{2}N_{2} c̄^{2}_{2 } → equation (2)

When the temperature of both gases is same, their mean kinetic energies per molecule will also be the same.

1/2 m_{1} c̄^{2}_{1} = 1/2 m_{2} c̄^{2}_{2}

Further,

m_{1} c̄^{2}_{1} = m_{2} c̄^{2}_{2 }_{ } → equation (3)

Divide both above equations (2) and (3), we get,

N_{1} = N_{2 }(Avogadro’s law)

Hence, equal volumes of all gases at the same temperature and pressure contain an equal number of molecules.

### Graham law of Diffusion

In 1848, a Scottish physical chemist, Thomas Graham formulated a relationship between the rate of diffusion or effusion with that of the molar mass of gases. It stated that the rate of diffusion or effusion of gases is inversely proportional to the square root of its mass density.

r ∝ √(1/d)

### Derivation of Graham’s law from KMT

Applying the kinetic equation,

PV = 1/3 mNc̄^{2}

If one mole of gas has Avogadro’s number of molecules N=N_{A}

PV = ⅓ mN_{A} c̄^{2}

As, M (molecular mass of the gas) = mN_{A}

PV = 1/3 Mc̄^{2}

Rearranging,

c̄^{2} = 3PV/M

Taking square root on both sides,

√c̄^{2} = √(3PV/M)

Now as, mass/volume = density

√c̄^{2} = √(3P/d)

where,

- d is the mass density of a gas

The root mean square velocity of the gas is proportional to the rate of diffusion of gas,

**√c̄ ^{2} ∝ r**

By relating the above equations, we get

r ∝ √3P/d

And if there is constant pressure,

r ∝ √(1/d) (Graham’s law)

## KMT explains the relation between K.E and T

As the Postulate 7 of kinetic molecular theory (KMT) suggests, the kinetic molecular theory is directly proportional to the absolute temperature.

As we know that, the kinetic equation of gases

PV =1/3 mNc̄^{2}

The average kinetic energy associated with one molecule of gas due to translatory motion is as follows.

E_{k} = ½ mc̄^{2}

We know from Boyle’s law that:

PV = ⅔ N ( ½ mc̄^{2})

By relating the above equations we get,

PV = ⅔ N.E_{k}

This equation gives an important view of the exact meaning of temperature. To understand one mole of gas,

N = N^{A}

PV = ⅔ N_{A }E_{k } → equation (3)

The ideal gas equation can be written for one mole as:

PV = RT_{ } → equation (4)

Comparing both equations (3) and (4), we get,

2/3 N_{A }E_{k }= RT

E_{k} = ( 3R / 2N_{A} ) T

Hence,

E_{k} ∝ T

(The kinetic energy of a gas is directly proportional to the absolute temperature of that gas)

## Key Takeaway(s)

- Kinetic molecular theory (KMT) explains all laws of gaseous thermodynamics such as Boyle’s law, Charles law, Avogadro’s law, and graham’s law.
- KMT allows us to thermodynamically explain the existence of solid, liquid, and gas states of matter.
- It allows us to explain the physical characteristics and phase changes between solids, liquids, and gases.

## Concepts Berg

**What is an example of kinetic molecular theory?**

The simplest example of the kinetic molecular theory is the diffusion of gases in space. The Brownian motion of gas particles is according to this theory as well.

**How does the kinetic theory explain gas pressure?**

The pressure of gas applied on the walls of the container is due to the collision of gas particles. The greater the speed of particles, the greater would be the collisions of particles, hence, more pressure will be exerted.

**What is the kinetic molecular theory based upon?**

The kinetic molecular theory is based upon experimental observations and evidence independent of the nature of gases.

**What are the applications of kinetic molecular theory?**

The kinetic molecular theory states that matter is always in continuous motion. Therefore it can be applied to study the phase changes in matter i.e. from solid to liquid then to the gas phase. Other applications include Brownian motion and diffusion etc.

**Is the kinetic molecular theory only applied to ideal gases?**

Yes, the kinetic molecular theory is applied to ideal gases only. In reality, there are some deviations.

**Does kinetic molecular theory violate thermodynamics?**

Kinetic theory provides the basic framework of thermodynamics. It does not violate thermodynamics.

**What is evaporation in context with kinetic theory?**

Evaporation is the continuous conversion of a liquid from the liquid phase to gas on the surface. All liquid particles are in certain kinetic energy that is explained by the Maxwell Boltzmann distribution curve.

**Who gave the kinetic theory of gases?**

Clausius proposed this theory in 1857. He got the basis of this idea from the molecular theory of gases presented by Bernoulli earlier in 1738. This theory led to the foundation of thermodynamics and scientists deduced gas laws from it. Extension to this theory was given by Maxwell and Boltzmann later on studied the kinetic aspects.

**Is the kinetic theory of gases valid today?**

Kinetic theory of gases is used to derive and explain the gas laws. All gas laws are obeyed by ideal gases. We applied these laws to real gases. So we can say that this theory is valid today when we are dealing with thermodynamic aspects on a macroscopic level of ideal cases.

**What is the kinetic gas equation?**

c_{rms} = √(3RT/M)

- c
_{rms}= root mean square velocity - M= molar mass
- T = temperature

This equation shows the direct relationship between temperature and the velocity of gases. Moreover, this equation can be conveniently used to explain all of the ideal gas laws.

**References**

- The Kinetic Molecular Theory
Purdue University, Indiana**(****)** An Introduction to the Kinetic Theory of Gases by James Jeans,

**Sir James Hopwood Jeans**(University of**Cambridge**) and (**Princeton University**)